Asset Location and Allocation with
Multiple Risky Assets
Ashraf Al Zaman∗
Krannert Graduate School of Management, Purdue University, IN
zamanaa@mgmt.purdue.edu
May 4, 2004
Abstract
Most working adults have access to a taxable brokerage account (TBA) and
a tax deferred retirement account (TDRA). According to the existing literature,
taxable bonds should be located in the TDRA, while equities should be located
in the TBA due to the tax treatments of these accounts. If borrowing is not
allowed mixed holdings can be optimal in either account but not in both simultaneously. But if borrowing is allowed all the wealth in the TDRA should be
∗
I am extremely grateful to Mike Cliff and John J. McConnell for their continuous guidance and
support. Thanks to John M. Barron and C. D. Aliprantis for their helpful comments. I would also
like to thank the seminar participants at Krannert Graduate School of Management for their valuable
suggestions. Thanks to Chester S. Spatt and Harold H. Zhang for their helpful suggestions and
comments. Comments of all the participants of the Asset Allocation and Mortality Conference at the
Fields Institute, Toronto, helped me in improving the layout and content of this work.
1
allocated to bonds. Unfortunately, the empirical findings are at odds with the
theoretical predictions as investors do hold both equities and bonds in both of the
accounts. This discrepancy is known as the asset location puzzle. In this paper,
we revisit the asset location issue by extending the model to include multiple
risky assets. This allows us to capture the interaction of portfolio diversification
and the tax timing option, features that are not captured in the existing models. We find that the correlation structure of the risky assets and the borrowing
constraints have substantial impact on asset location decisions. But this impact
is sensitive to various levels of borrowing constraints. We also find that asset
location and allocation decisions are sensitive to the Sharpe ratios of the risky
assets, and the size of the retirement account. Consistent with the existing empirical findings, we also observe interesting mix of bonds and equities in both of
the accounts with borrowing constraints and with reasonable relaxation of the
borrowing constraints. Furthermore, we document the impact of borrowing constraints, and show that the assumption of unlimited borrowing is not innocuous.
Failing to incorporate institutional restrictions on borrowing might have led to
the theoretical inconsistencies.
2
1
Introduction
One of the most important decisions any working adult has to make pertains to the
composition of his or her investment portfolio. In making investment decisions, she
is faced with two major questions. The first question relates to asset allocation—how
should she allocate her investment resources among various risky and risk free assets.
The second question relates to asset location—given the availability of both taxable
and tax-deferred savings accounts, how should she locate the risky and risk free assets
in these two accounts.
There is a large literature addressing the asset allocation issue. But the issue of
asset location has received less attention in the literature. In most of the portfolio
optimization papers, taxes are nonexistent. However, taxes can have important effects
on portfolio choice in real life. With differential tax treatments between the TBAs
and the TDRAs, one asset (e.g., Microsoft stock) can now have multiple after-tax
payoff structures. Consequently, the presence of tax-advantaged accounts influences
the standard asset location problem. Investors not only must choose how much money
to put in each asset, but also where those assets should be held.
Due to certain tax advantages in various tax deferred retirement accounts (TDRAs)
as compared to taxable brokerage accounts (TBAs), taxes can play an important role
in portfolio decisions. In a taxable account, investors have incentives to realize capital
losses yet defer capital gains. This incentives imply that investors hold assets that
have done well and sell those assets that have done poorly. Over time, following this
strategy will lead to a poorly diversified portfolio. Thus, there is a trade off between
the utilizing the tax timing option and maintaining a well-diversified portfolio.
3
In the last few years, several researchers have tried to address the portfolio location and allocation problem jointly with limited success. Results presented by the
researchers are mixed, and models highly stylized and incomplete.1 Dammon, Spatt,
and Zhang (2004), hereafter DSZ (2004), conclude that the optimal location for the
taxable bonds is the TDRA, and the optimal location for equities is the TBA given
that borrowing is not restricted. Using an arbitrage argument, Huang (2000) obtains
similar results with deterministic liquidity shocks. The intuition driving their results
is that assets with high yields (e.g., bonds) expose investors to larger tax burdens and
should therefore be held in the tax deferred account. Moreover, by holding stocks in
the TBA, the investor retains the tax timing option and can meet her liquidity needs
without any penalty. In contrast to DSZ (2004) and Huang (2000), Shoven (1999)
finds that hosting equity mutual funds in their tax-differed accounts may be optimal
as opposed to hosting taxable corporate bonds in that location.2
The findings of DSZ (2004) and Huang (2000) are at odds with much of the empirical
evidence on asset location. Barber and Odean (2001) find mixed holding of stocks and
bonds in the taxable accounts of the customers of a retail brokerage. The DSZ (2004)
framework is unable to explain the mixed holdings, and the authors call this empirical
finding as the asset location puzzle in DSZ (2004). These discrepancies between the
empirical and theoretical findings imply either that the models being considered are
incomplete, or an appreciable number of investors are making suboptimal investment
decisions.
1
see Barber and Odean (2001)
2
DSZ (2004) document some specific conditions under which Shoven (1999) conclusions are feasible
within their framework.
4
In this paper, we revisit the asset location and allocation problem. The main
difference in our analysis as compared to DSZ (2004) is that we consider multiple
risky assets as opposed to a single risky asset. This seemingly simple extension can
have important consequences on the optimal portfolio due to the incorporation of the
potential of diversification. In particular, it is often the case that investors will choose
to put a mix of equities and bonds in both the TBA and the TDRA for viable parameter
values and borrowing restrictions. Moreover, one of the key contributions of our paper is
the analysis of the relationship between location decision and the correlation structure
of the risky assets.
Why does adding a second risky asset change the optimal portfolio so much? The
answer is fairly simple. With a single risky asset, there is no concern about diversification. With multiple risky assets, the desire to defer gains and realize losses leaves an
investor with a poorly diversified portfolio. By shifting some equities into the TDRA,
the investor can freely trade these assets to maintain a diversified portfolio without
incurring the tax consequences. The goal of the investor is to reduce the tax burden
on the financial asset holdings as much as possible while at the same time maintaining
a portfolio as diversified as possible. In part, these results are also due to more realistic treatment of constraints individual investors face on short sales and borrowing.
Moreover, by considering multiple risky assets, we increase the value of the tax timing
option, which is understated when a single market index is considered.
When investors are restricted from borrowing and short selling, they are unable to
realize the full benefit of their tax timing options. We explore the impact that these
restrictions can have on the optimal portfolio. From the casual observation of invest-
5
ment opportunities, institutional regulations, and industry practices, it is perceived
that borrowing and short selling are not as readily available services for the average
investors as have been presented in the literature. Moreover, whenever borrowing is
allowed, the interest rates for borrowing and lending are not the same. Hence the
liquidity needs cannot be met as easily with borrowing or selling of equities.
With multiple risky assets in the portfolio, any liquidation decision must take into
account of the cost basis of the existing equities. If an investor needs a certain amount
of cash, she may meet this need by liquidating any equity. But ignoring the cost
basis of the equities may lead to extra tax burden. In terms of liquidation, high cost
basis equities are preferred for liquidation over the low cost basis equities. Sometimes
high return equities may be liquidated due to their high cost basis. So the cross-basis
effects due to holding of multiple risky assets become very important with regards to
liquidation for certain shocks. Liquidation may also be motivated by diversification
concerns. For example, if a portfolio becomes heavily weighted in one risky asset, in
order to maintain a desired level of diversification one may need to liquidate some
of her position, and the liquidation decision is not independent of the consideration
of the cost basis of other assets in the portfolio. Sometimes portfolio rebalancing
for diversification purpose may reduce the value of the tax timing option. To get
insight into all these issues we extend the existing models as discussed below and make
qualitative conclusions.
To capture the diversification or portfolio rebalancing issues in a more pronounced
manner, we consider two risky assets or equities, and a risk free taxable bond in our
model. We capture the cross-basis effect by allowing for various parameterizations of
6
the basis of the risky assets. To capture the institutional restrictions such as borrowing
and short selling, we consider three different cases, 1) constraints on both borrowing
and short sales, 2) constraints on short sales but not on borrowing, and 3) constraints
on none. We also consider various correlation structures in order to have specific insight
into diversification. Limited analysis of bequest motive, retirement contribution limit,
and optimal size of the retirement account is also conducted.
Our model predicts mixed holdings of equities and bonds in either the TBA or the
TDRA under borrowing constraint when the equity prices are independent to each
other. The level of the mix is dependent upon the retirement wealth ratio.3 But
mixed holdings are not observed in both accounts simultaneously. When retirement
wealth ratio is high, and borrowing is not allowed some equities may spill over to the
TDRA due to the investor’s desire for higher (overall) risk exposure. These findings
are consistent with DSZ (2004). The mix in the TDRA tends to disappear as the
borrowing constraint is gradually relaxed. When the borrowing constraint is relaxed,
we still observe mixed holdings of equities and bonds in the TDRA if the retirement
wealth ratio is high. The level of borrowing required to have only bonds in the TDRA
is exorbitant in some cases. Some of these results are sensitive to parameter values such
as retirement wealth ratio. For example, to have all of the TDRA wealth in bonds, an
investor with average retirement wealth ratio (y = 0.4) needs to borrow up to 100% of
her TBA wealth whereas an investor with high retirement wealth ratio (y = 0.7) needs
to borrow up to 175% of her TBA wealth. With borrowing constraints investors simply
3
Retirement wealth ratio is measured as the fraction of the total wealth that is held in the retirement
account. More detailed description is given in the following sections, and the technical description is
given in the Appendix.
7
can not achieve their desired risk exposure by holding only bonds in the TDRA. Due
to lack of borrowing ability investors can not balance their long position in bonds in
the TDRA with a comparable position in equities in the TBA. We observe relatively
similar results when the short sale constraint is relaxed.
The correlation structure of the risky assets is a key determinant of the location
decision. Under borrowing constraint, negative correlations generally lead to all equty
allocation in the TDRA. But when borrowing constraint is relaxed up to 100%, mixed
holding is observed in the TDRA for correlation coefficient of −0.4 and lower. Moreover,
we notice that with multiple risky assets, the Sharpe ratios may be a key determinant
of bond holding in the TBA under borrowing constraint. The bequest motive does not
affect asset location decisions in substantial manner during the working age but does
affect the investor’s consumption and equity holding during the retirement age. Retirement contribution limits do not affect location decisions but mildly affect allocation
decisions in some cases. Furthermore, we observe a potential relationship between the
optimal size of the retirement account and the borrowing constraint.
The rest of the paper proceeds as follows. In section 2, we discuss the existing
models and their results, in section 3 we present the model and the methodology of
our investigation, in section 4 we present the results, and we conclude in section 5.
2
Literature Review
As has been noted in Leland (2000), portfolio management entails two decisions. The
investor should first specify the target or ideal portfolio strategy, which determines the
8
desired proportions of investment in different classes of assets such as stocks and bonds.
The literature provides a rigorous framework for static and dynamic asset allocation
in the absence of trading costs and other market frictions to facilitate the first decision
(e.g., Markowitz (1952) and Merton (1971)). The second decision is how to implement
the desired strategy. Issues to be considered in this implementation step include taxes,
borrowing costs, market structure, financial regulations, trading costs, and costs related
to other market frictions. Most of the papers in the portfolio optimization area consider
the two-decision process separately, abstracting away from the dynamic interaction of
the decisions and their outcomes. Our paper addresses both of the decisions and their
interactions.
Constantinides (1983) introduced the idea of a tax timing option and pioneered the
study of optimal investment and liquidation policy under capital gains taxes. According
to the prescription in Constantinides (1983), investors should realize capital losses
immediately and defer capital gains realization indefinitely. An important assumption
is that there are no restrictions on short sales. In that case, the optimal portfolio choice
is separable from the liquidation policy, since “overexposure” to an asset for tax timing
reasons can be “undone” by offsetting short positions. A substantial amount of work
has built around this basic foundation. Constantinides (1984) shows an application
of the findings of Constantinides (1983). Dammon and Spatt (1996) derive the value
of tax timing option with the key assumption that investors can circumvent the wash
sale rules.4 For further references on this strand of literature, interested readers may
4
Current IRS rules restricts wash sales. The scope and enforcement of the rules has increased over
time.
9
consult Dammon and Spatt (1996).
Most of the existing literature on asset location and allocation focuses on an investor
who has access to one risky asset and one risk free asset, a TBA, and a TDRA. The
investor needs to decide in which assets she should invest, and where those assets should
be hosted. The tax environment plays a major role in this decision process. The general
result suggests that bonds should be held in the TDRA and equities should be held
in the TBA. The following paragraphs explain the intuition of this analysis, using the
arbitrage arguments presented by DSZ (2004).
Investors receive dividends from the investments in stocks and interest payments
from their investments in bonds. Sale of stocks results in capital gains or losses. In the
TBA, either type of income is taxed at the rate of the investor’s personal income tax
rate (τd ) while capital gains are taxed at the capital gains tax rate (τg ). Suppose that
the rate of interest (r) is greater than the dividend yield (di ) of asset i, so for assets
held in the taxable account, the tax burden on the income from bonds is greater than
that on the dividend income from stocks (rτd > di τd ). Taxes on capital gains (or losses)
on equities in the TBA are due upon liquidation not upon accrual, so investors can
avoid paying taxes by continuing to hold the equities. In contrast, the tax on income
(interest and dividend payments) or on capital gains from equities can be deferred into
the future (e.g., until retirement) if the assets are held in the TDRA. Moreover, due to
the basis step-up provision, all the taxes may be forgiven at the death of the investor.
As a result of these features of the accounts, the choice of asset location affects the
after-tax payoff structure of the assets. These in turn may affect the asset allocation
decision.
10
Assume that an investor is shifting one of her dollars from stock i to bonds in her
tax deferred account, which is offset by a shift of xi dollars from bonds to that stock
in her taxable account. Denote the change in the net cash flow at the end of the year
by ∆Ci then we have the following
∆Ci = r − [(1 + gei )(1 + di) − 1] + xi {[(1 + gei )(1 + di (1 − τd )) − gei τg − 1] − r(1 − τd )} (1)
where gei is the random capital gain. Taking the partial derivative of ∆Ci with respect
to gei provides an expression, which does not change in value due to any change in gei .
Setting this expression to zero and solving for xi gives
xi =
1 + di
1 + di (1 − τd ) − τg
(2)
which can then be plugged back into (1), and after rearranging the terms we obtain
∆Ci = xi
"
(r − di )(τd − τg )
1 + di
#
(3)
Note that the expression of ∆Ci is free of gei, and it represents a risk-free, after-
tax cash flow that an investor can generate by simple relocation of the assets without
incurring any cost. This is the basic intuition of the arbitrage argument.
Given that the term τd − τg in (3) is positive,5 it is clear that the sign of ∆Ci
depends on the sign of the term r − di . Since the interest on a bond is generally
higher than the dividend yield on a stock, this spread is typically positive. When
∆Ci is positive, the investor is strictly better off holding the bond in the tax deferred
account and holding stock i in the taxable account. Note that
5
In the case where τd = τg , the location of the assets does not matter.
11
∂∆Ci
∂di
< 0, hence ∆Ci
is monotonically decreasing in di . Thus, for every asset where r − di > 0 the investor
allocates bonds in the tax-deferred account, and stocks in the taxable account.
In general, the investor will always prefer to allocate her entire tax- deferred wealth
to the asset with the highest yield, and allocate other assets in the taxable account.
This basic principle can be used to decide on the location of any financial assets including various stock funds, mutual funds, taxable and non-taxable bonds. In DSZ
(2004) further results are obtained by imposing restriction on borrowing. With this
constraint, the investor first shifts her tax-deferred wealth into the asset with highest yield until no more of this shifting is possible. Then she allocates her remaining
tax-deferred wealth to the asset with the next highest yield, and so on. The process continues with successively lower yielding assets until the investor’s tax-deferred
wealth is allocated completely. Then she makes offsetting adjustments in the taxable
account for the desired level of risk exposure. But sometimes it may not be possible to
make the offsetting adjustments due to the borrowing and short-sale restrictions. This
may restrict her from having all bonds in the TDRA. Thus, with borrowing and short
sale constraints, the investor may hold a mix of taxable bonds and equities in her tax
deferred account, but only if the investor holds all equities in her taxable account.
Shoven and Sialm (2001) also address the asset location and allocation issue but
they do so using considerably different modeling techniques. They use a continuous
time approach but ignore the consumption/savings decision. They consider a broader
menu of assets, including mutual funds and tax-exempt municipal bonds. According
to them the preferred asset location is determined primarily by the tax rates facing
the asset returns, and assets with the high tax rates should be allocated in the tax-
12
deferred account. Taxable bonds should be held in the tax-deferred account whereas
tax-exempt bonds should be held in taxable account. Stocks can be held in either
account depending on the tax-efficiency of the stock or stock portfolios.
The prescription of the existing theoretical models for investors to put taxable
bonds in the tax deferred account, and equities in the taxable account stands in sharp
contrast with observed investor behavior. DSZ (2004) considers this inconsistency
as the asset location puzzle. Using the data from the Survey of Consumer Finances,
Bergstresser and Poterba (2001) show that many households hold stocks in their taxdeferred account but not in their taxable account. This suggests that substantial group
of households do not follow a “bond first in the tax-deferred account” asset location
strategy. In a similar study, Barber and Odean (2001) find mixed holdings of stocks and
bonds in both taxable and tax-deferred accounts (data sources are retail and discount
brokerages).
One possible explanation for the discrepancies between theoretical predictions and
empirical facts are the modeling assumptions. Under realistic restrictions on borrowing
and short selling, mixed holdings are feasible. As has been noted in DSZ (2004),
if investors are not allowed to borrow they may maintain mixed holdings. But the
prediction of all bond allocation in the TDRA with unlimited borrowing is not viable in
reality because of institutional restrictions on borrowing. For most individual investors
borrowing and short selling constraints are clearly binding. Diversification concerns
may also lead to mixed holding if the correlation structure of equities is amicable to
diversification. With good opportunities for diversification, mixed holdings may be
observed even without borrowing and short sale constraints. Superior value out of
13
diversification may outweigh the value of tax timing option. Moreover, the retirement
wealth ratio of the investor in general may matter in portfolio choice. For example,
consider an investor with high retirement wealth ratio and a preference for higher risk
exposure who is facing borrowing and short selling constraints. Even though only
bonds are optimally located in the TDRA, equities may spill over to the TDRA simply
because her TBA account is too small, and she is unable to borrow to achieve her
desired risk exposure through borrowing in the TBA.
Amromin (2001) offers an explanation for the empirical findings. His explanation
is based on liquidity and the accessibility restrictions on assets in the TDRA. Investors
would like to have access to their investment funds in the event of a negative income
shock. Due to restrictions on accessing funds in the TDRA, investors have a preference
for having some bonds in the TBA.
In this paper, we introduce a dynamic model with multiple risky assets to consider
the diversification issue with regards to asset location and allocation decisions. We
also consider the impact of various restrictions such as borrowing and short selling
constraints on asset location and allocation. To the best of our knowledge, no one has
considered the asset location and allocation issue with multiple equity assets yet. Most
of the modeling has been done by using only one risky asset (generally interpreted as a
well-diversified market equity portfolio) and a risk free bond. The absence of multiple
risky assets restricts us from observing the interaction of the value of diversification
and tax timing option. If the goal of the investor is to hold a well-diversified portfolio
while maximizing the after-tax value of her investment within the tax and institutional
constraints then the existing models are certainly limited in many ways. We capture
14
the impact of diversification by having multiple risky assets in our model. We also
consider the robustness of our prediction to bequest motives. Our extension helps in
explaining some of the inconsistencies between theoretical and empirical findings.
3
The Model
We consider a 20-year-old investor who works for 50 years, then lives in retirement
for another 30 years before dying at age 100. The investor tries to smoothen her
consumption, and wants to leave some of her wealth as a bequest. Here we assume
that the investor and her descendant have the same preference over consumption and
formulate the bequest as an H-period annuity, i.e. whatever amount the investor leaves
for her descendant at her death is invested in an annuity, and the descendant receives
the annuity payments over H years. Thus the investor implicitly chooses the value of
the bequest then she maximizes her utility over consumption.
The investor has access to two equities, and a taxable bond maturing in one year.
Each equity price follows a binomial price process. The assets can be held in a TBA
and/or a TDRA. The equity price processes are assumed to be independent of each
other.6 During her working years, she receives labor or non-financial income of L in
each period. She contributes a fixed proportion of her labor income (α) to her TDRA.
But she is not allowed to withdraw any amount out of this account till retirement.7
6
We consider the correlated cases in latter sections.
7
We make this simplifying assumption for the ease of our analysis. Moreover, given the penalty
one has to pay in order to withdraw money from the tax-deferred account before retirement, it is
unlikely that an average investor would want to do that in regular circumstances.
15
The investor seeks to maximize her utility of lifetime consumption by choosing seven
control variables at each point in time. The control variables are the allocation toward
consumption (Ct ), the number of shares of equity one and two held after rebalancing
the TBA holdings (n1t and n2t ), amount of bonds (Bt )held in the TBA, the fraction of
retirement wealth held in equities (θ1t and θ2t ), and the fraction of retirement wealth
held in bonds (θ3t ) after rebalancing the TDRA . The rate of inflation is denoted by i.
The formal optimization problem (with the constraints to be discussed later) is as
follows8
max
Ct ,n1t ,n2t ,Bt ,θ1t ,θ2t ,θ3t

E0 
TX
−1
t=0
β tU
Ct
(1 + i)t
!
+ βT
H
X
j=1
βj U
!
AH WT 
(1 + i)T
(4)
The objective function has two components. The first component represents the utility
of the investor, defined over real consumption, throughout the working and retirement
years (discounted at a rate of β). The second component represents the utility of the
descendant out of the consumption of the bequest over H years (discounted at a rate
of β). The ending period wealth is W T , and AH is the H-period annuity factor. So
AH W T is the annual consumption of the heir. The expectation is taken over the whole
expression due to the random processes followed by the equity prices.
The maximization of the objective function, Equation (4), is subject to a number of
constraints. We describe these conceptually here and refer the reader to the Appendix
for a formal presentation.9
C1 Total wealth each period is the sum of the wealth in the the TBA (see C2) and
8
To maintain the clarity of the exposition, we describe most of the notation and definitions in the
Appendix.
9
In the Appendix, the constraints C1 - C9 is presented in order as Equations 6 - 14.
16
the TDRA (see C3) after subtracting the non-capital gains tax liabilities.
C2 Wealth in the TBA each period is the after tax-labor income plus the value of the
holdings of the equities and bond after paying taxes on interest and dividends
(before rebalancing the portfolio).
C3 Wealth in the TDRA each period is the prior-period ending balance times the
TDRA portfolio gross return (no taxes are deducted).
C4 Consumption each period is the residual left after subtracting from total wealth
(C1) the value of both investment accounts and any capital gains taxes in the
TBA.
C5 The amount of money in the TDRA at the start of each period until retirement
is the prior balance plus a fraction (α) of the labor income.
C6 The amount of money in the TDRA at the start of each period in retirement is
the prior balance less a withdrawal used for subsistence.
C7 Consumption must be non-negative in every period. Short selling is not allowed
in the TBA.10 Borrowing is not allowed in the TBA.11 And investors hold nonnegative amount of each asset in the TDRA.
C8 The investor must liquidate positions at death.
C9 Average cost basis is used.
10
We relax this constraint latter on to analyze the consequences of having this constraint.
11
We relax this constraint latter on to analyze the consequences of having this constraint.
17
We use the power form of the utility function with constant relative risk aversion
coefficient γ > 0.
U(Ct ) =









ln(Ct ) if γ = 1
Ct1−γ
1−γ
otherwise
This form of the utility function is chosen due to the convenience of being able to make
conclusions that are independent of wealth level. This simplifies the analysis since we
can ignore wealth as a state variable.
In the dynamic optimization problem, the investor observes a vector of state varih
i
∗
∗
ables, Xt = P1t , P1t−1
, P2t , P2t−1
, n1t−1 , n2t−1 , Wt , Yt , Lt . These variables contain in-
formation known at the time the investor chooses the control variables. As a modeling
convenience, we follow DSZ (2004) and normalize the vector of state variables to obtain
xt = [s1t , s2t , p∗1t−1 , p∗2t−1 , yt ]
12
. The symbols skt , p∗kt−1 , and yt represent prior holding
of stock k, k = 1, 2, as a fraction of the wealth in the TBA (Wt ), basis-price ratio
at time t, and wealth in the TDRA as a fraction of the beginning of the period total
wealth (W t ) before trading at time t, respectively. The consumption wealth ratio is
ct .13 The variables bt , f1t , and f2t track the fraction of wealth in the TBA allocated
to the risk free taxable bond, equity 1, and equity 2, respectively. The fraction of the
TDRA wealth allocated to equity 1, equity 2, and the taxable bond are θ1t , θ2t , and
θ3t , respectively. We assume that labor income or non-financial income is a constant
fraction (l) of the total wealth during the working years and is zero in retirement.
12
This allows transformation of the optimization problem into a more tractable form. Please consult
the Appendix for the normalization procedure and the reformulated optimization problem.
13
ct is the fraction of the total wealth allocated for consumption at time t.
18
3.1
Limitations and Potential Extensions
Certain limitations of the model need to be recognized. In our model, we have the labor
or non-financial income as a constant fraction of the total wealth i.e. it does not follow
any random process of its own. This assumption allows us to use the homogeneity of
the objective function in reducing the dynamic optimization problem (we discuss this
in the Appendix). Given the findings of Amromin (2001) and Huang (2000), we believe
that introducing labor income shocks would strengthen our conclusions as it introduces
other incentive to hold bond in the TBA. We use the average cost basis calculation
rather than exact cost basis. In an exact cost basis calculation, the changing path
of the cost basis is taken into account for each transaction for a given asset, and the
cost basis is updated accordingly. The loss due to this simplification should be minor.
DeMiguel and Uppal (2003) reports that the certainty equivalent loss from using the
average tax basis instead of exact tax basis is less than 0.2% for problems with five
periods and is less than 0.5% for problems with ten periods. Even though the loss
is increasing in the time periods considered, the impact of it would be minor on our
qualitative conclusions due to the magnitude of the optimized values of the relevant
variables that we use to make our conclusions. Finally, we have a definite death date
as opposed to a stochastic one. Overall, we believe that our main results would not
change in any drastic manner due to any of the concerns we have mentioned here.
19
4
Results from Numerical Optimization
We solve the investor’s dynamic optimization problem numerically. The Appendix
contains some technical details and a summary follows below.
To parameterize the model, we make the following assumptions.14 The agent is
20 at the beginning. Then she works for 50 years. Then lives in retirement for 30
years before dying at age 100. The equities have mean returns (µ) of 9% and 13%,
respectively. Corresponding standard deviations (σ) are 20% and 30%. Dividend rate
for both of the stocks is 2%, and the interest rate is 6%. The tax rates are 36% for
ordinary income and 20% for capital gains. The inflation rate is 3.5%.
The risk aversion parameter (γ) is set at 3. We set the fraction of total wealth earned
as labor or non-financial income (l) at 15% at any given time during the working age.
The portion of labor income saved in the retirement account (α) is 20%. To capture
the bequest motives we vary the bequest horizon parameter H. But for most of the
analysis, we set H at 20. In other words, in most of the cases we consider an investor
who intends to leave a bequest amount that would allow her heir to have a fixed level
of consumption for 20 years.
The numerical optimization is done over a grid of possible values for the state vector.
We created a 5 × 5 × 5 × 5 × 8 grid for the state space and used this grid for each of
the 80 periods for the optimization (s1t ∈ [0.01, 1], s2t ∈ [0.01, 1], p∗1t−1 ∈ [0.01, 1.2],
p∗2t−1 ∈ [0.01, 1.2], and yt ∈ [0.1, .8]). We have tried all parameterizations within the
stated range but we report only a few interesting parametric outcomes. We solve the
14
We closely follow the parameter values from DSZ (2004) to make the comparison of the qualitative
conclusions easier. DSZ (2004) documents viable reasons for selecting certain parameter values.
20
dynamic optimization problem by using backward induction and linear interpolation.
We start with the terminal value of the value function of the investor. Then using the
backward induction, we figure out in order to have that terminal value which asset
location and allocation decisions the investor needs to make. We do this for each state
vector in the grid. Once we cover the whole state space, we go backward one more
period and repeat the same procedure. We continue this till we get to the initial period.
In the figures low value of parameters s1t and s2t (e.g., 0.1) indicate lower level
of prior equity holdings, and high value of parameters s1t and s2t (e.g., 0.7) indicate
higher level of prior equity holdings. If the basis-price (p∗1t−1 or p∗2t−1 ) ratio is above
one, we have built in capital losses. When the basis-price ratio is below one, we have a
built in capital gains, and when it is equal to one there are no capital gains or losses.
Here we focus specifically on parameter values that provide new insights related
to the existing literature. Existing theoretical findings fail to explain the mixed holdings observed in the empirical data and ascribe this to suboptimal decision making
in the part of the investors. Here we present our work in a way that may help us in
understanding some of the reasons for these theoretical inconsistencies. The results
in this section are organized as follows. First, we discuss the investor’s asset location
and allocation decisions. Then we consider the issue of bequest motives. We discuss
the retirement contribution limit, and its impact on location and allocation decision in
the following subsection. In the following part of this section, we discuss the optimal
size of the retirement account. At the end, we consider the investor’s consumption
choices. For the ease of presentation, we start with the base portfolio and vary parameter values of the base portfolio in order to get other specific portfolios or cases. For
21
the base case we consider a portfolio that is heavily weighted in equities, both equities
are equally weighted, and there are no built in capital gains or losses, (s1t = s2t = .4,
p∗1t−1 = p∗2t−1 = 1), please refer to Table 1 for details. To focus on the process of mixed
holdings of stocks and bonds, we report combined equity allocation throughout this
presentation.
22
4.1
Asset location and allocation
We observe interesting mix of bonds and equities in both of the accounts, the TBA
and the TDRA, with and without borrowing constraint relaxation. The location and
allocation decisions are sensitive to specific parameter values such as the basis-price
ratios of the equities, the retirement wealth levels, Sharpe ratios of the equities, correlation structure of the equities, and the trading restrictions. We present the results in
the following way. We first present asset location and allocation decisions for the base
portfolio under various constraints. Then we present the asset location scenarios under
various correlation structures. The relationship between retirement wealth ratio (y)
and the TDRA equity holding under various levels of borrowing restriction is discussed
at the end of this section.
Consider the base portfolio described earlier, see Table 1 for the set of parameter
values. In this scenario, with borrowing and short sale constraints investors do not
hold any bonds in the TBA irrespective of the retirement wealth ratio. We observe
mixed holdings in the TDRA only for individuals with higher y values. In panel (a)
and (b) of Figure 1, we present the allocations of the total wealth of an investor with
y = 0.1 in the TBA and the TDRA. Preferred location for equities is the TBA and
for bonds it is the TDRA. Only around 10% of the total wealth is allocated to bonds
but equity allocation is almost 80%, and equity allocation in the TBA increases during
the retirement age. No bond is held in the TBA. Total allocation toward consumption
and equities in the TBA is little less than 90% of the total wealth, see the line labeled
Total. On the contrary, for investor with y = .8 allocations, presented in panel (c)
and (d) of Figure 1, are different. There is a mix of bond and equity allocation in
23
the TDRA. Almost 65% of the total wealth is held in equities in the TDRA and this
allocation is mildly declining in age. But less than 10% of the total wealth is allocated
to the equities in the TBA during the working age. Equity holding is increasing in age
in the TBA, the maximum is approximately 30%. Overall equity allocation seems to
be negatively related to the retirement wealth ratio.
Now we consider wealth allocations across retirement wealth ratios (y) over time
within the TBA and the TDRA. In this scenario, with borrowing and short sale constraints investors hold no bonds in the TBA irrespective of the retirement wealth ratio,
see Figure 2 (c). Here equity allocation in the TBA is decreasing in retirement wealth
ratio during the working years. The allocation to equity varies between 100 − 40%.
That is investors invest 100 − 40% of their TBA wealth in Equities depending on their
retirement wealth ratios. But equity holding is increasing in the retirement wealth
ratio during the retirement age, see Figure 2 (a). For investors with high retirement
wealth ratios, it is optimal to allocate most of their funds to equities in the TDRA for
tax deferred growth whereas the allocation in the TBA is more suitable for immediate
liquidity needs. Holding equities in the TBA allows them to meet their liquidity needs,
capture the growth in capital gains, and leave the door open for exercising the tax timing option. As opposed to investors with higher retirement wealth ratios, for investors
with lower retirement wealth ratios liquidity needs may be much stronger. So they tend
to hold more stock in the TBA, and try to meet their liquidity needs as well as try
to capitalize on the tax timing option. During retirement age bequest motive pushes
the equity holding further up. This push may be due to the potential of capitalizing
on tax the basis step-up or due to the availability of retirement funds that allows the
24
investor to spare more for investments. Notice in Figure 2 (a) for the investors with
high retirement wealth ratios, the allocation towards equities during retirement years
may go above 100%. The allocation is measured as a fraction of the wealth in the
TBA at any given time. During the retirement age, the non-financial or labor income
is zero but the investor receives returns from investments in the TBA, and a fraction
of the TDRA wealth is also available from the retirement distributions. Given the bequest motive and reduction in gain from tax deferral during retirement, investors with
high retirement wealth ratios may invest the extra money they receive from the TDRA
distributions (amount available after consumption) in the TBA. Hence the relative investment amount in equity in the TBA may go beyond 100% even with the borrowing
constraint for individuals with high y. The proportion of overall equity holding in the
TBA tend to remain the same but the composition of the portfolio is affected by the
tax basis and prior equity holdings. Given the opportunity of diversification, investors
may opt for more equities given better risk adjusted return.
Sometimes location of bond in the TBA may be motivated by the inability to borrow
to meet the liquidity needs. We notice an interesting relationship between the Sharpe
ratio and bond holding. Up to certain threshold Sharpe ratio, bond allocation in the
TBA is negatively related to the Sharpe ratios. We present this relation graphically in
Figure 5 (a).
In the TDRA, bond holding is decreasing in retirement wealth ratio across time.
For low retirement wealth ratio almost 100% of the retirement wealth is allocated
to bonds. But it declines substantially over retirement wealth ratio, and it is below
50% for investors with high retirement wealth ratios, see Figure 2 (d). The equity
25
allocation is increasing in retirement wealth ratio in the TDRA, see Figure 2 (b). Here
we obtain results similar to DSZ (2004). We have only equity allocation in the TBA
and mix allocation in the TDRA. But mixed holdings in both accounts are not observed
simultaneously.
When borrowing constraint is relaxed, we observe interesting changes in the asset
location and allocation patterns. Investors with low retirement wealth ratio still prefer
higher overall equity exposure as compared to the investors with high retirement wealth
ratio. Borrowing is higher during the working age as opposed to the retirement age.
But investors with high wealth ratio increase their equity holdings in the TBA, and
decrease their equity holdings in the TDRA. For investors with low retirement wealth
ratio, asset location decision does not change but the allocation to the equities increase
in the TBA. When investors are allowed to borrow up to 100% of their TBA wealth,
on average an investor with y = 0.1 borrows around 13.40% of her total wealth during
her working age, and borrows around 10.70% of her total wealth during her retirement
age. Equity holding increases in both part of the investors life, during working age it
stands at 93.73%, and during retirement age it stands at 97.56%. But bond holding is
relatively low, which stands at 10% of the total wealth. And no bonds are held in the
TDRA. These results are summarized in Table 2. The Magnitude column indicates
the fraction of the wealth within a particular account allocated to a particular asset
class or consumption. And the rest of the columns contain the frequencies of holdings
of assets of particular asset class in different accounts over the investor’s lifetime. The
two bottom rows present mean allocations for working age and retirement age.
Investors with higher retirement wealth make interesting location and allocation
26
decisions when they are allowed to borrow up to 100% of their TBA wealth. As opposed
to the individuals with y = 0.1, investors with y = 0.8 borrow aggressively in the TBA,
and increase their equity exposure. Interestingly, they reduce their exposure to equities
in the TDRA, and increase it in the TBA. Whereas equity exposure decreases, bond
exposure increases. On average equity allocation stands at 73.96% during the working
age, and at 82.40% during the retirement age. Equity allocations in the TBA and the
TDRA during the working age are 28.78% and 45.18%, respectively. In retirement age
these allocations are 42.09% and 40.31%, respectively. Investors borrow aggressively,
and borrow the maximum amount which is comparable to 20% of their total wealth.
These results and holding frequencies are tabulated in Table 3
Now we consider wealth allocations across retirement wealth ratios (y) over time
within the TBA and the TDRA. Investors hold no bonds in the TBA irrespective of
their retirement wealth ratios.15 Borrowing level is low for the investors with low retirement wealth ratios but investors with high retirement wealth ratios borrow heavily,
see Figure 3 (c). The relaxed borrowing constraint is binding for investors with higher
retirement wealth ratio, y = 0.5 − 0.8. Notice that for investors with lower retirement
wealth ratio, y = 0.1 − 0.5, the constraint is not binding. Even though these investors
could borrow more, they do not borrow to their maximum capacity.
When the borrowing constraint is relaxed, investors borrow heavily to capitalize
on higher returns on the equities. The equity holding shifts upward, and the equity
15
Throughout the presentation, when we mention borrowing is not restricted we mean limited
relaxation of the borrowing constraint. For the base case analysis, we allow the investor to borrow
up to 100% of her wealth in the TBA. Unless otherwise stated, whenever we mention that there is no
borrowing constraint, it should be understood that we are allowing the investor to borrow 100%.
27
holding may range from 100 − 250% of the wealth in the TBA depending on y and age.
But the allocation pattern during working age and retirement age remains very similar
to the constrained case, see Figure 3 (a). Given the relaxed borrowing constraint,
investors with high y borrow heavily in the TBA to capitalize on higher returns from
the equities. But the investors with lower level of retirement wealth are not as aggressive
in borrowing (for them the borrowing constraint is not binding), they borrow money
to hold more equity in the TBA but do not hold any equity in the TDRA. Their main
allocation of funds in the TDRA is in bonds. By relaxing short sale constraints we
obtain similar qualitative conclusions.
Now we consider various correlation structures of the equities using the base portfolio parameters. We consider equities with correlation coefficients (ρ) of −1, −0.6,
−0.4, −0.2, 0, 0.2, 0.4, 0.6, and 1. When the equities are perfectly positively correlated (ρ = 1) the equities do not serve much in terms of diversification. But close to
60% of the wealth in the TBA is held in equities before and after borrowing constraint
relaxation to capitalize on the risk adjusted returns and the tax timing option. But
as the magnitude of the correlation declines and the sign becomes negative the equity
holding increases in the TBA. Interesting insights are obtained with regards to the
TDRA. The location decision is influenced by the correlation structure of the equities.
With negative correlation the value of diversification is high so investors always hold
100% in equities in the TDRA. But if ρ = 0 i.e. if the equity price processes are independent of each other, the equity holding is lower with borrowing constraints. When
investors are allowed to borrow as much as 100% of their TBA wealth, they hold no
equities in the TDRA if the correlation coefficient is positive (if ρ = 0.6 or ρ = 1). But
28
if the correlation coefficient is negative all the retirement wealth is allocated towards
equities. By relaxing the borrowing constraint further it is possible to have all in bonds
in the TDRA but the level of borrowing is too high to be feasible. But when the equities are perfectly negatively correlated (ρ = −1), it is not possible to have bonds in
the TDRA even with high level of borrowing. The diversification benefit is simply too
high. The diversification value outweighs the value of the tax timing option. In panel
(a) and (b) of Figure 4, we present the allocations for equities with correlation coefficients (ρ) of −0.4, −0.2, 0, 0.2, 0.4 when borrowing is not allowed. In panel (c) and
(d), similar allocations are presented when borrowing constraint is relaxed up to 100%.
With borrowing constraint, it is always optimal to have 100% of the retirement wealth
in equities when correlation coefficient are negative. But the equity allocation is lower
when the equity returns are independent or positively correlated. When borrowing
constraint is relaxed, equity location preference changes dramatically. For correlation
coefficients of −0.2, 0, and 0.2 it is never optimal to host any equities in the TDRA,
100% of the TDRA wealth goes to bonds. Only for correlation coefficient of −0.4, it
is optimal to hold around 20% of the TDRA wealth in equities.
One of the key determinants of bond location in the TDRA is the borrowing constraint. We document the interaction of retirement wealth ratio (y) and borrowing
constraints in determining the bond holding in the TDRA for some specific parameter
values. In panel (a) and (b) of Figure 6, we consider two scenarios. In panel (a), we
have an investor with y = 0.4. This investor holds a mix of bonds and equities in the
TDRA. But when she is allowed to borrow 100% of her TBA wealth, she holds only
bonds in the TDRA. For making the point more clear, we also have an intermediate
29
case where we allow the investor to borrow only 25% of her TBA wealth. This shows
the dependence of the investor’s location preferences on borrowing ability. The level
of borrowing that is required to have all in bonds in the TDRA also depends on the
retirement wealth ratio this can be observed in panel (b) where y = 0.7. Here the
investor need to have much more borrowing ability in order to have total preference
for bond in the TDRA. The investor needs to be allowed to borrow up to 175% of
her TBA wealth to have all in bonds in her TDRA, which may deem unreasonable
given the institutional restrictions. We have tried other scenarios where one equity got
appreciated and other lost value; even in those scenarios we obtain similar results.
4.2
Bequest motive
The observations made above are mostly robust to changes in bequest motive, especially
during the working years. The most noticeable difference that we observe is in the
allocation for consumption, and equity holdings. We consider three specifications for
this purpose with H = 5, 20, and 25. The investor is allowed to borrow up to 100%
of her TBA wealth. The consumption level is around 10% of the total wealth during
most of the working age but in the retirement age the consumption allocation changes.
For low bequest motive, H = 5, we observe consumption increases dramatically over
time whereas for higher bequest motive, H = 20 consumption is decreasing over time
(see Figure 5 (b)). The growth in consumption for low bequest motive, H = 5, is
pronounced in retirement age. Even with low bequest motive consumption allocation
seems to be within the range of 10 − 15% of the total wealth. As the bequest motive
increases consumption in the retirement age decreases sharply with age. When bequest
30
motive is low, the investor derives utility mostly out of her own consumption, hence
she makes her consumption allocation decision accordingly. These conclusions seem
to hold for variations in initial equity holdings. The magnitude of various allocations
vary over bequest motives but the location decisions are qualitatively very similar in
the working age. Variations in allocation strategy do arise in the retirement age. To
provide some intuition, we consider the equity and bond allocation in the TBA for
an investor with average retirement wealth, y = .4, holding the base portfolio. The
allocations are presented graphically in panel (c) and (d) of Figure 5. In panel (c),
the equity allocation in the TBA is considered. Equity holding in the retirement age
declines for low bequest motive but increases for higher bequest motive. An investor
with low bequest motive is focused in maximizing her own consumption within her
life, so the value of the tax timing option or the benefit of the step-up clause is not
as relevant to her. But for a person with higher bequest motive all these issues are
important. Holding equity allows her to capitalize on the tax timing option and on
the potential growth in capital gains. Panel (d) presents the bond allocation, and the
bond allocation is uniform for all bequest motives. Heavy borrowing in the working
age and reduction in borrowing in the retirement age. Investors tend to borrow heavily
to capture the benefit of the higher returns of equity and on the deferral option. But
in the retirement age investors start to dissave and borrowing is not as much needed to
meet the liquidity need for consumption, or asset purchases. Equity holding is almost
zero in the TDRA. All the wealth in the TDRA is allocated toward bond for growth
with tax deferral.
31
4.3
Retirement contribution limit
Retirement contribution limit does not affect the asset location decision but within
the accounts it affects the allocation decisions. To compare the changes in utility, we
use certainty equivalent consumption. In the base case, retirement contribution level
(α) is set at 20%. In order to get insight into the issue, we keep all the parameter
values for the base case the same except for α. We vary the value of α to observe its
impact on location and allocation decisions. With borrowing constraint, we observe
minor reallocation of assets in the TBA, and all the changes in allocation take place
during the working years. With higher value of α, investors tend to reduce their equity
holding in the TBA by 0 − 5%. Investors with the lower y values benefit more from
the relaxation of the contribution limit. But the gain in utility from the increase in the
value of α is not strictly monotonically increasing, see Table 4. In column two of Table
4, we report the percentage changes in the certainty equivalent consumption due to
the increase in the retirement contribution level from 5% to 20%. We observe that the
investor with y = 0.1 benefits most and the investor with y = 0.7 benefits least from
the increase in α. We do a similar analysis without borrowing constraint in this case,
the order of benefit received is maintained but the magnitude of the benefit increases,
see column three of Table 4. Similar results are reported for the increase in α from
20% to 30% in column four and five of Table 4.
4.4
Optimal size of the retirement account
In our analysis, we have the retirement wealth ratio (y) as a state variable. This can be
thought of as a proxy for the size of the retirement account. Any investor would want
32
to maximize her utility by allocating her entire wealth in the two accounts optimally.
Here we present some analysis on this issue. We consider the base case parameters,
and vary the y values and try to observe the impact of these variations on utility, and
measure these variations in terms of certainty equivalent consumption. In panel (c)
of Figure 6, we present the case when borrowing is not allowed. The investors with
higher y values are better off overall which is quite intuitive. But the interesting case
is presented in panel (d) where the investors are allowed to borrow up to 100% of their
TBA wealth. We observe that optimality is not linearly dependent on the size of the
retirement wealth ratio. Here the investor with y = 0.6 is better off than the investor
with y = 0.8. So it is interesting to notice that there may be a key role of borrowing
constraint in the determination of the optimal size of the retirement account.
4.5
Consumption
Consumption is relatively smooth across age and retirement wealth ratio. The level of
consumption varies according to age, bequest motive, level of retirement wealth ratio,
correlation structure of the risky assets, and restrictions on trading. Depending on
various parameter values, the level of consumption (ct ) expressed as the fraction of the
total wealth allocated toward consumption may vary between 3.5 − 13%. Our findings
regarding consumption is mostly consistent with the existing findings in the qualitative
sense.
We simulate for three different cases due to trading restrictions. In the first case,
we don’t allow any borrowing and short selling, in the second case we allow borrowing,
and in the third case we allow both borrowing and short selling. When borrowing
33
and short selling are allowed the investment opportunity set for the investors expands
which allows the investors to consume at a higher level. We present the impact of the
restrictions graphically for the investors with average retirement wealth ratio, y = 0.4,
in Figure 7 (a). In this figure, C1 represents the level of consumption with restrictions
on both borrowing and short selling, C2 represents the level of consumption when
only borrowing is allowed, and curve C3 represents the level of consumption under no
restrictions. Level of consumption is lowest in C1 but higher in C2 and C3. Consumption is decreasing in age irrespective of trading restrictions. Notice further that the
gain in consumption from the relaxation of short sale constraint is not as substantive
as it is in the case of borrowing constraint relaxation. The sharp decline in ct during
the working age is due to the fixed contribution that the investors make towards the
TDRA out of their non-financial income. This fixed contribution increases the level
of the total wealth of the investors over time. Since consumption level is relatively
constant and the total wealth is growing over time, the value of ct declines sharply
during the working years. But the decline in ct during the retirement age is relatively
minor due to the absence of the fixed contribution of part of the non-financial income
towards the TDRA. Rather than contributing to the TDRA, during the retirement age
investors withdraw wealth from the TDRA for consumption purposes. The investors
try to balance their consumption with their bequest motives. But ct increases dramatically for the investors with low bequest motive. Since the investors are not leaving
much for their heirs, they spend more of their total wealth for consumption. This
intuition is graphically presented in Figure 5 (b).
In general, consumption is decreasing in age and mildly increasing in retirement
34
wealth ratio. Figure 7 (b) presents consumption across retirement wealth ratio over
time under borrowing constraint. Here consumption is smooth across time and retirement wealth ratio. When borrowing constraint is relaxed the consumption surface
shifts upward but the relative consumption patterns are maintained for all retirement
wealth ratios (see Figure 7 (c)) . Retirement wealth ratio affects consumption level but
the impact is minor, in Figure 7 (d) we consider two retirement wealth levels to make
this point. Here we consider investors who are not allowed to borrow. In Figure 7 (d),
we present consumption profiles of two investors. The first investor has low retirement
wealth ratio, y = .1, and the second investor has high retirement wealth ratio, y = .8.
As can be observed from the figure, consumption level is higher for the investor with
higher retirement wealth ratio.
Within any single period, consumption is relatively smooth across basis price ratio
and prior equity holding. In any single period consumption level is dependent on the
basis-price ratios of the equities and equity holdings. If equity holding level for either
equities is high, the determinant of the consumption level is the basis-price ratio. If
the basis-price ratio is low i.e. there is substantial built in capital gain then the
consumption level would be low. But if there is built in capital loss consumption level
may be higher. If there are built in capital losses, in the beginning of the period the
investors will capitalize on the tax timing option i.e. they would claim the capital losses,
and either consume more or rebalance their portfolios with the proceeds. Further, if
prior equity holding is high, consumption level is higher in basis-price ratio. The
consumption level of an investor with high equity holding with built in capital gains
is relatively low compare to the investor with low equity holding with built in capital
35
gains. The fraction of wealth that an investor with high level of prior equity holding
with built in capital gains spends for consumption would certainly be smaller because
of her high net wealth. But this outcome changes when basis-price ratio is greater than
one.
5
Conclusion
Most of the modeling framework in the existing literature of asset location and allocation promote the view that by having larger coupon rates as compared to the dividend
yield rates, an investor is much more exposed to tax burden by holding bonds in the
taxable investment account (TBA) as opposed to holding it in the tax differed account
(TDRA). Moreover, by holding stocks in the taxable account an investor can capitalize
on the tax timing option, and meet her liquidity needs without any penalty. But the
limitations of the existing models are substantial that begs careful reformulation and
augmentation. In this paper, we introduce the issue of diversification in this discussion. And we find that correlation structure of the risky assets in the portfolio can be
a key determinant of location decision. In many cases, we obtain results very similar
to existing results. Specifically, we observe that DSZ (2004) conclusions under borrowing constraints do hold for viable parameter values with multiple risky assets. But
we observe that when investors are allowed to borrow the existing conclusions would
hold in some cases only under demanding assumptions on borrowing. Under borrowing constraints, negatively correlated assets would always be optimally allocated in
the TDRA indicating the superiority of the value of diversification over the value of
36
tax timing option. Moreover, when the risky assets are perfectly negatively correlated
hardly any assumption on borrowing would buttress the conclusions in DSZ (2004).
We find interesting mix of equities and bonds in the TBA and in the TDRA with the
borrowing constraint and with limited (viable) relaxation of the borrowing constraint,
which is consistent with the empirical findings. Our analysis of bequest motives show
that location decisions are robust to bequest motives. So in general, we can conclude
that an average investor should take into account of the correlation structure of the
risky assets in the portfolio along with borrowing constraints, retirement wealth ratio,
basis-price ratio, and Sharpe ratios of the risky assets in making asset location and allocation decisions. Retirement contribution limit do not affect the location decision. We
have touched upon the optimal size of the retirement account, but we believe a more
thorough investigation would be fruitful in this area. Our limited analysis indicates
that there may be an important relationship between the optimal size of the retirement
account and the investor’s borrowing ability. Addressing some of the limitations and
concerns that we have mentioned in the main body of the paper would certainly facilitate our understanding of asset location and allocation further. Most of the existing
empirical data is aggregate level data. But more primary, and individual investor level
data would certainly strengthen the empirical conclusions, and would allow us to test
the quality of the theoretical models. For now our conclusions and analysis suffices as
evidence in reconciling some of the concerns regarding the asset location puzzle.
37
6
Appendix
The investor solves the following optimization problem for asset location and allocation.16
max
Ct ,n1t ,n2t ,Bt ,θ1t ,θ2t ,θ3t

E0 
TX
−1
t=0
t
βU
Ct
(1 + i)t
!
+β
T
H
X
j
β U
j=1
!
AH WT 
(1 + i)T
(5)
such that
Wt = Lt (1 − τd ) +
2
X
Wt = Wt + Yt (1 − τd ), t = 0, . . . , T
(6)
nkt−1 [1 + (1 − τd )dk ]Pkt + Bt−1 [1 + (1 − τd )r], t = 0, . . . , T
(7)
k=1
Yt =
r
Wt−1
"
2
X
#
θkt−1 (1 + gkt )(1 + dk ) + θ3t−1 (1 + r) , t = 0, . . . , T
k=1
Ct = Wt − τg
2
X
k=1
Gkt −
2
X
nkt Pkt − Bt − Wtr (1 − τd ), t = 0, . . . , T − 1
(9)
k=1
Wtr = Yt + αLt , t = 0, . . . , T r − 1
16
(8)
(10)
The objective function discribed in Equation 4 (in page-16) is the same as the objective function
discribed in Equation 5. The constraints C1-C9 are presented as Equations (6)-(14), respectively.
46
Wtr = Yt (1 − ht ), t = T r, . . . , T − 1
(11)
Ct ≥ 0, nkt ≥ 0, Bkt ≥ 0, 0 ≤ θit ≤ 1, t = 0, . . . , T − 1, i = 1, 2, 3.
(12)
nkT = 0, BT = 0, WTr = 0
(13)
∗
Pkt
=





∗
nkt−1 Pkt−1
+max(nkt −nkt−1 ,0)Pkt
nkt−1 +max(nkt −nkt−1 ,0)




∗
if Pkt−1
< Pkt
(14)
Pkt if Pkt−1 ≥ Pkt
n
o
∗
∗
∗
Gkt = I(Pkt−1
> Pkt )nkt−1 + [1 − I(Pkt−1
> Pkt )]max(nkt−1 − nkt , 0) (Pkt − Pkt−1
)(15)
Notations:
Ct = nominal consumption at time t
Bt = amount invested in bonds in the TBA at time t
Wt = wealth in taxable account after payment of the ordinary income taxes but prior
to the payment of capital gains taxes at time t
Wt = total wealth at time t
Wtr = wealth in the TDRA after contribution or withdrawal at time t
Yt = pretax wealth in the tax-deferred account before contribution or withdrawal at
time t
47
Lt = pretax non-financial or labor income in period t
αLt = contribution to the retirement account from the pretax non-financial income at
time t
ht Yt = withdrawal from the retirement account at time t
Pkt = price of stock k at time t
nkt = number of the shares of stock k held in the TBA
r = nominal risk free interest rate
d = nominal dividend yield
gkt = nominal pre-tax capital gain return from stock k in period t
Gkt = total realized capital gain from stock k at time t
τd = income tax rate
τg = capital gains tax rate
H = number of years for which the investor wants to leave funds for her descendant
or bequest horizon
i = inflation rate
Inflation adjusted annuity factor:
AH =
r ∗ (1 + r ∗ )H
(1 + r ∗ )H − 1
r ∗ = [(1 − τd )r − i]/(1 + i)
We reformulate the constrained optimization problem as a dynamic optimization problem. Then the value function of the dynamic optimization problem at time t, Vt (Xt ),
is a function of the vector of the state variables, Xt , at time t.
48
(
)
Ct
Vt (Xt ) = max U(
) + βEt [Vt+1 (Xt+1 )]
(1 + i)t
which is subject to the constraints listed in Equations 6 - 14.
h
∗
∗
Xt = P1t , P1t−1
, P2t , P2t−1
, n1t−1 , n2t−1 , Wt , Yt , Lt
i
We normalize the value function and introduce some new variables to simplify
the problem further. Let vt =
f1t =
n1t P1t
,
Wt
f2t =
n2t P2t
,
Wt
bt =
V (Xt )
,
[W /(1+i)t ]1−γ
Bt
,
Wt
ct =
Ct
,
W
l =
yt =
Lt
,
Wt
s1t =
Yt (1−τd )
,
W
wtr =
n1t−1 P1t
,
Wt
Wtr
,
W
s2t =
n2t−1 P2t
,
Wt
δkt = Gkt /Wt , and
∗
p∗kt−1 = Pkt−1
/Pt .
After the normalization, we deal with five state variables, beginning of the period
equity 1 proportion or holding in the TBA (s1t ), beginning of the period equity 2
proportion or holding in the TBA (s2t ), basis-price ratio of equity 1 (p∗1t−1 ), basis-price
ratio of equity 2 (p∗2t−1 ), and retirement wealth ratio (yt ) at time t, respectively. We
redefine the vector of state variables as xt = [s1t , s2t , p∗1t−1 , p∗2t−1 , yt ]. Note that l denotes
the constant fraction of the total wealth that comes from the labor or non-financial
income, Lt . Retirement withdrawal rate is denoted by ht , and it is calculated by using
the life expectancy table provided by IRS (ht is the inverse of life expectancy at time
t).
For control variables, we have ct , bt , f1t , f2t , θ1t , θ2t , and θ3t , consumption-wealth
ratio, fraction of the TBA wealth allocated to bonds after trading, fraction of the TBA
wealth allocated to equity 1 after trading, fraction of the TBA wealth allocated to
equity 2 after trading, fraction of the TDRA wealth allocated to equity 1, fraction of
49
the TDRA wealth allocated to equity 2, and fraction of the TDRA wealth allocated
to bonds, respectively, at time t. Using the new notations, we can reformulate the
constraints as follows.
(
ct = 1 − τg δt (1 − yt ) − (1 − yt ) bt +
Rt+1 =
bt [1 + (1 − τd )r] +
2
X
fkt − wtr (1 − τd )
k=1
P2
r
Rt+1
= θ3t (1 + r) +
k=1 fkt [1 + (1
P
bt + 2k=1 fkt
2
X
)
− τd )dk ](1 + gkt+1 )
θkt (1 + dk )(1 + gkt+1 )
(16)
(17)
(18)
k=1
Equations (17) and (18) can be considered as the gross returns in the TBA and
the TDRA. With the new notation and reorganization, we have the following reduced
dynamic optimization problem
(
)
ct1−γ
vt (xt ) = max
+ βEt [vt+1 (xt+1 )wt+1 1−γ ]
1−γ
(19)
such that
wt+1
2
r
X
Rt+1
1 − yt
Rt+1
=
(bt +
fkt )
+
1+i
1 − l(1 − τd )
1+i
k=1
wtr =
!
wtr (1 − τd )
, t = 0, · · · , T − 1(20)
1 − l(1 − τd )
yt
+ αl, t = 0, · · · , T r − 1
1 − τd
50
(21)
wtr =
yt
(1 − ht ), t = T r, · · · , T − 1
1 − τd
ct ≥ 0, fkt ≥ 0, 1 ≥ θit ≥ 0, i = 1, 2
(22)
(23)
At the terminal date T , the value function take the value vT .
1−γ
β(1 − β H )AH
vT =
(1 − β)(1 − γ)
(24)
We create a 5 × 5 × 5 × 5 × 8 grid for the state space such that s1t ∈ [0.01, 1],
s2t ∈ [0.01, 1], p∗1t−1 ∈ [0.01, 1.2], p∗2t−1 ∈ [0.01, 1.2], and yt ∈ [0.1, .8]. We optimize
over this state space for each of the 80 periods. We solve the dynamic optimization
problem by using backward induction and linear interpolation. We keep the state space
relatively small to economize on computation time and difficulty. But grid points are
selected in such a way that allows us to obtain qualitative inferences about the location
allocation decisions for most of the interesting cases.
51
References
[1] Amromin, G., (2001), “Portfolio Allocation Choices in Taxable and Tax-Deferred
Accounts: An Empirical Analysis of Tax Efficiency,” Unpublished manuscript,
University of Chicago.
[2] Barber, B. M., and Odean, T., (2001), “Are Individual Investors Tax Savvy? Asset
location Evidence from Retail and Discount Brokerage Accounts,” Conference
Paper, Stanford University.
[3] Bergstresser, D., and Poterba, J., (2001), “Asset Allocation and Asset Location
Decisions: Evidence from the Survey of Consumer Finances,” Conference Paper,
Stanford University.
[4] Constantinides, G. (1983), “Capital Market Equilibrium with Personal Taxes,”
Econometrica, 51, 611-636.
[5] Constantinides, G. (1984), “Optimal Stock Trading with Personal Taxes: Implecations for Prices and the Abnormal January Returns,” Journal of Financial
Economics, 13, 65-89.
[6] Dammon, R. M., and Spatt, C.S., (1996), “The Optimal Trading and Pricing of
Securities with Asymmetric Capital Gain Taxes and Transaction Costs,” Review
of Financial Studies, 9, 921-952.
[7] Dammon, R. M., and Spatt, C.S., and Zhang, H. H., (2002), “Optimal Asset Location and Allocation with Taxable and Tax-Deferred Investing,” Working Paper,
Carnegie Mellon University.
52
[8] Dammon, R. M., and Spatt, C.S., and Zhang, H. H., (2003), “Optimal Asset
Location and Allocation with Taxable and Tax-Deferred Investing,” Forthcoming
in Journal of Finance.
[9] DeMiguel, A. V., and Uppal, R., (2003), “Portfolio Investment with the Exact
Tax Basis via Nonlinear Programming,” Working Paper, London Business School.
[10] Epstein, L., and Zin, S., (1989), “Substitution, Risk Aversion, and the temporal
Behavior of Consumption and Asset returns: A Theoretical Framework,” Econometrica, 57, 937-968.
[11] Epstein, L., and Zin, S., (1991), “Substitution, Risk Aversion, and the temporal
Behavior of Consumption and Asset returns: An Empirical Investigation,” Journal
of Political Economy, 99,263-286.
[12] Huang, J., (2000), “Taxable or Tax-Deferred Account? Portfolio Decision with
Multiple Investment Goals,” Unpublished manuscript, MIT.
[13] Garlappi, L., Naik, V., and Slive, J., (2001), “Portfolio Selection with Multiple
Risky Assets and Capital Gains Taxes,” Working Paper, The University of British
Columbia.
[14] Leland, H., (2000). “Optimal Portfolio Implementation with Transaction Costs
and Capital Gains Taxes,” University of California at Barkley. Hass School of
Business Working Paper.
[15] Magill, M., and Constantinides, G., (1976), “Portfolio Selection with Transaction
Costs,” Journal of Economic Theory, 13, 245-263.
53
[16] Markowitz, H., (1952), “Portfolio Selection,” Journal of Finance, 7, 77-91.
[17] Shoven, J., (1999), “The Location and Allocation of Assets in Pension and Conventional Savings Accounts,”National Bureau of Economic Research Working Paper.
[18] Shoven, J., and Sialm, C., (2001), “Asset Location for Tax-Deferred and Conventional Savings Accounts,”Unpublished manuscript, Stanford University.
54
Table 1: Table of Parameter Values for the Base Portfolio
Parameters
Base portfolio
Basis-price ratio of asset 1 (p∗1t−1 )
1.0
Basis-price ratio of asset 2 (p∗2t−1 )
1.0
Prior holdings in equity 1 (s1t )
0.4
Prior holdings in equity 2 (s2t )
0.4
Mean return of equity 1 (µ1 )
9%
Standard deviation of equity 1 (σ1 )
20%
Mean return of equity 2 (µ2 )
13%
Standard deviation of equity 2 (σ2 )
30%
Correlation coefficient(ρ)
0
Bequest horizon (H)
20
Retirement wealth ratio (y)
0.4
Interest rate (r)
6%
Inflation rate (i)
3.5%
Ordinary income tax rate (τd )
36%
Capital gains tax rate (τg )
20%
Risk aversion parameter (γ)
3
Fraction of total wealth earned as labor income (l)
15%
Fraction of labor income contributed to the TDRA (α)
20%
55
Table 2: Asset Holding Frequency and Wealth Allocations for an Investor with Low
Retirement Wealth Ratio (y = .1)
TBA
Magnitude
Consumption
Equity
< −90%
0.00
0.00
< −40%
0.00
< −10%
TDRA
Equity
Bond
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
98.75
0.00
0.00
< 0%
0.00
0.00
100
0.00
0.00
> 0%
100
100
0.00
0.00
100
> 5%
57.50
100
0.00
0.00
100
> 10%
22.50
100
0.00
0.00
0.00
> 50%
0.00
100
0.00
0.00
0.00
Mean wealth allocations 8.75%
Bond
92.73% −13.40% 0.00%
10%
97.56% −10.70% 0.00%
10%
(Working age)
Mean wealth allocations 3.92%
(Retirement age)
This table shows the frequencies of various asset holdings, and their magnitudes in
different accounts over the lifetime of the investor. The two bottom rows show the
mean wealth allocations in these accounts. This table is based on the values of the
base portfolio parameters. The investor is allowed to borrow up to 100% of her TBA
wealth. The Magnitude column indicates the fraction of the wealth within a particular
account allocated to a particular asset class or consumption. And the rest of the
columns contain the frequencies of holdings of assets of particular asset class in different
56 show how much of the wealth in each
accounts over her lifetime. The bottom rows
account is allocated for different asset classes and consumption.
Table 3: Asset Holding Frequency and Wealth Allocations for an Investor with High
Retirement Wealth Ratio (y = .8)
TBA
TDRA
Magnitude
Consumption
Equity
Bond
Equity
< −90%
0.00
0.00
0.00
0.00
0.00
< −40%
0.00
0.00
0.00
0.00
0.00
< −10%
0.00
0.00
100
0.00
0.00
< 0%
0.00
0.00
100
0.00
0.00
> 0%
100
100
0.00
100
100
> 5%
60
100
0.00
100
100
> 10%
32.50
100
0.00
100
100
> 50%
0.00
0.00
0.00
0.00
0.00
Mean wealth allocations 9.30%
Bond
28.77% −20% 45.18% 34.82%
(Working age)
Mean wealth allocations 4.14%
42.09% −20% 40.31% 39.69%
(Retirement age)
This table shows the frequency of various asset holdings, and their magnitudes in
different accounts over the lifetime of the investor. The two bottom rows show the
mean wealth allocations in these accounts. This table is based on the values of the
base portfolio parameters. The investor is allowed to borrow up to 100% of her TBA
wealth. The Magnitude column indicates the fraction of the wealth within a particular
account allocated to a particular asset class or consumption. And the rest of the
columns contain the frequencies of holdings of assets of particular asset class in different
57 show how much of the wealth in each
accounts over her lifetime. The bottom rows
account is allocated for different asset classes and consumption.
Table 4: Changes in Utility Due to the Variations in the Retirement Contribution Level
∆v5−20% with ∆v5−20% with-
∆v20−30%
borrowing
out borrowing
with borrow- out borrowing
constraint
constraint
ing constraint
constraint
.1 0.141%
0.265%
0.086%
0.176%
.4 0.090%
0.254%
0.060%
0.169%
.7 0.088%
0.086%
0.059%
0.057%
∆v20−30% with-
y
This table presents the average percentage changes in the certainty equivalent consumption level (during the working years) due to the changes in the retirement contribution
limit. The changes are measured by measuring the changes in the certainty equivalent
consumption due to the changes in the retirement contribution level parameter (α).
The retirement wealth ratios (y) are listed in the first column. The second column
lists changes in the certainty equivalent consumption due to the change in the contribution level from 5% to 20% when borrowing is not allowed, and the third column lists
changes in the certainty equivalent consumption due to the change in the contribution
level from 5% to 20% when the investor can borrow up to 100% of her TBA wealth.
Similarly, columns four and five list the changes in certainty equivalent consumption
due to the changes in the contribution level from 20% to 30%, with and without the
borrowing constraint, respectively.
58
(a)
(b)
Consumption
Equity
Bond
Total
80
60
Allocations in the TDRA (%)
Allocations in the TBA (%)
100
40
20
0
−20
0
20
40
Time
60
Equity
Bond
100
80
60
40
20
0
80
0
20
(c)
80
60
80
70
Consumption
Equity
Bond
Total
30
Allocations in the TDRA (%)
Allocations in the TBA (%)
60
(d)
40
20
10
0
−10
40
Time
0
20
40
Time
60
50
40
30
20
10
80
Equity
Bond
60
0
20
40
Time
Figure 1: Allocations within the TBA and the TDRA
Parameter values are of the base portfolio. The investor is not allowed to borrow. Panel
(a) and (b) show the wealth allocations of an investor with low retirement wealth ratio
(y = 0.1) within the TBA and the TDRA. Panel (c) and (d) show the wealth allocations
of an investor with high retirement wealth ratio (y = 0.8) within the TBA and the
TDRA. In panel (a) and (c) lines referred to as Total show the total allocations for
consumption, bonds, and equities in the TBA.
59
Equity in the TDRA (%)
(a)
200
Equity in the TBA (%)
(b)
100
100
50
0
80
0
80
60
60
40
40
20
Time
0.4
0.2
0
0.6
0.8
20
Time
0
Retirement wealth ratio (y)
Bond in the TDRA (%)
(c)
0.2
0.8
0.6
0.4
Retirement wealth ratio (y)
(d)
Bond in the TBA (%)
100
1
0.5
50
0
80
60
0
40
50
Time
0
0.2
0.4
0.6
0.8
Time
Retirement wealth ratio (y)
20
0
0.2
0.8
0.6
0.4
Retirement wealth ratio (y)
Figure 2: Asset Allocation and Location Under Borrowing and Short Sale Constraints
Parameter values are of the base portfolio. The investor faces both borrowing and short
sale constraints. (a) Combined equity holding in the TBA across time and retirement
wealth ratio. (b) Combined equity holding in the TDRA across time and retirement
wealth ratio. (c) Bond holding in the TBA across time and retirement wealth ratio.
(d) Bond holding in the TDRA across time and retirement wealth ratio.
60
Equity in the TBA (%)
300
Equity in the TDRA (%)
(b)
(a)
200
100
80
60
40
20
0
80
60
60
40
40
20
0
Time
0.4
0.2
0.6
0.8
20
Time
0
0.2
Bond in the TBA (%)
(c)
Bond in the TDRA (%)
Retirement wealth ratio (y)
0.8
0.6
0.4
Retirement wealth ratio (y)
(d)
100
0
−50
80
60
40
80
−100
80
60
60
40
40
20
Time
0
0.2
0.8
0.6
0.4
Retirement wealth ratio (y)
20
Time
0
0.2
0.8
0.6
0.4
Retirement wealth ratio (y)
Figure 3: Asset Allocation and Location when Borrowing Constraint is Relaxed
Parameter values are of the base portfolio. The investor is allowed to borrow up to
100% of her TBA wealth. (a) Combined equity holding in the TBA across time and
retirement wealth ratio. (b) Combined equity holding in the TDRA across time and
retirement wealth ratio. (c) Bond holding in the TBA across time and retirement
wealth ratio. (d) Bond holding in the TDRA across time and retirement wealth ratio.
61
(a)
= corr(−0.2)
= corr(0)
= corr(0.2)
80
= corr(−0.4)
= corr(0)
= corr(0.4)
100
Equity in the TDRA (%)
100
Equity in the TDRA (%)
(b)
60
40
20
0
80
60
40
20
0
0
20
40
Time
60
80
0
20
40
Time
(c)
60
80
= corr(−0.4)
= corr(0)
= corr(0.4)
100
Equity in the TDRA (%)
Equity in the TDRA (%)
80
80
(d)
= corr(−0.2)
= corr(0)
= corr(0.2)
100
60
60
40
20
0
80
60
40
20
0
0
20
40
Time
60
80
0
20
40
Time
Figure 4: Correlation Structure and Equity Holdings in the TDRA
Parameter values are of the base portfolio. The plots present the impact of correlation
structure on the level of equity holding in the TDRA. Panel (a) and (b) present the
cases when borrowing is not allowed, and panel (c) and (d) present the cases when the
investor is allowed to borrow up to 100% of her TBA wealth. Panel (a) presents three
curves indicating the levels of equity holding in the TDRA depending on correlation
coefficeints of −0.2, 0, and 0.2. Panel (b) presents three curves indicating the levels of
equity holding in the TDRA depending on correlation coefficeints of −0.4, 0, and 0.4.
62
(a)
(b)
14
H=5
H = 20
H = 25
30
80
20
60
10
Consumption (%)
Bond holding in the TBA (%)
12
40
0
0.1
10
8
6
4
2
20
0.15
0.2
0
0.25
Sharpe ratio
0
Time
0
20
(c)
80
20
H=5
H = 20
H = 25
180
Bond in the TBA (%)
Equity in the TBA (%)
60
(d)
200
160
140
120
100
80
40
Time
0
20
40
Time
60
0
−20
−40
−60
−80
80
H=5
H = 20
H = 25
0
20
40
Time
60
80
Figure 5: Sharpe Ratio and Bequest Motive
Parameter values are of the base portfolio. (a) Relationship between bond holding in
the TBA and Sharpe ratio of the second risky asset. A vector of standard deviations,
σ2 ∈ {25%, 30%, 35%, 40%, 45%, 50%, 55%, 60%, 65%}, is used to generate the Sharpe
ratios. Then the Sharpe ratios are plotted against the bond holding in the TBA when
borrowing is not allowed. In panel (b), (c), and (d), we consider an investor who is
allowed to borrow upto 100% of her TBA wealth. Panel (b) presents her consumption
allocations under various bequest motives. Panel (c) presents her combined equity
holding in the TBA for various bequest motives. Panel (d) presents her bond holding
in the TBA for various bequest motives.
63
(b)
100
Bond holding in the TDRA
Bond holding in the TDRA
(a)
80
60
40
20
0% borrowing
25% borrowing
100% borrowing
0
0
20
40
Time
60
100
80
60
40
20
0% borrowing
100% borrowing
175% borrowing
0
80
0
20
40
Time
0.035
0.03
0.025
0.02
y = 0.2
y = 0.4
y = 0.6
y = 0.8
0.015
0.01
0.005
0
20
40
Time
80
(d)
0.04
Certainty equivalent consumption
Certainty equivalent consumption
(c)
60
60
80
0.04
0.035
0.03
0.025
0.02
y = 0.2
y = 0.4
y = 0.6
y = 0.8
0.015
0.01
0.005
0
20
40
Time
60
80
Figure 6: Interaction between Borrowing Restrictions and Retirement Wealth Ratio,
and Optimal Size of the TDRA
Parameter values are of the base portfolio. The plots in panel (a) and (b) depict bond
holdings in the TDRA for three levels of borrowing constraints given the retirement
wealth ratio (y). Panel (a) and (b)presents the cases for y = .4 and y = .7, respectively.
The curves represent the cases when no borrowing is allowed (0% borrowing), when
the investor is allowed to borrow up to 25% of her TBA wealth, and when the investor
is allowed to borrow up to 100% of her TBA wealth. Panel (c) and (d) present various
certainty equivalent consumption levels due to the variations in the retirement wealth
ratios (y). Here value of y is a proxy for the size of the TDRA. Panel (c) presents the
case when the investor is restricted from borrowing. Panel (d) presents the case when
the investor is allowed to borrow upto 100%
64 of her TBA wealth.
(a)
(b)
C1
C2
C3
10
Consumption (%)
Consumption (%)
12
8
6
20
10
0
0
20
4
2
40
0.2
0.4
0.6
Retirement wealth ratio (y)
60
0
20
40
Time
60
Time 80
80
(c)
0.8
(d)
20
Consumption (%)
Consumption (%)
12
10
0
0
20
y = 0.1
y = 0.8
10
8
6
4
40
0.2
60
Time 80
0.8
0.4
0.6
Retirement wealth ratio (y)
2
0
20
40
Time
60
80
Figure 7: Interaction of Consumption, Retirement Wealth Ratio and Borrowing Constraints
Parameter values are of the base portfolio. When the investor is allowed to borrow
or borrowing constraint is relaxed, she is allowed to borrow up to 100% of her TBA
wealth. (a) Consumption under various restrictions over time. C1 represents consumption under both short sale and borrowing constraints, C2 represents consumption with
only short sale constraint, and C3 represents consumption without borrowing or short
sale constraints. (b) Allocation for consumption with borrowing and short sale constraints across time and retirement wealth ratio. (c) Allocation for consumption when
borrowing constraint is relaxed but short sale constraints is not relaxed. (d) Allocation
for consumption for individuals with low (y = .1) and high (y = .8) retirement wealth
ratio facing borrowing constraint.
65